PhD research topic - Adriana Garroni
Variational methods and applications to material science
Several models in Material Science, such as inelastic behaviours of materials (fractures, plasticity, damage), are sources of very deep mathematical questions. They exhibit complex phenomena, such as formation of microstructure, which are crucial for the understanding of material properties. This is the case of phase transitions for smart materials or of the interaction and evolution of material defects. Those phenomena can be treated by means of the tools of Calculus of Variations and Geometric measure Theory. They are often characterised by energies that might be non smooth or non convex and their analysis requires the use of suitable functional spaces (such as the BV space or the space of rectifiable currents). The complexity of the energies involved can be resolved by means of an asymptotic multi-scale analysis in terms of the so called Gamma-convergence (a variational convergence which guarantees the convergence of the corresponding minimum problems), and the use of relaxation techniques.
Some of the problems that can be consider in this context are the following:
- Derivation of variational models for topological defects in crystal. Analysis and evolution of defects.
- Analysis of crystal plasticity at different scales: Discrete defects in 3D, formation of grain boundaries, macroscopic model for plasticity via homogenization.
- Analysis of 2D models for monolayer materials (such as graphene), including the modeling of defects
- Models for fracture, damage and fatigue.
 Braides, A., Gamma-convergence for beginners. Oxford Lecture Series in Mathematics and its Applications, 22. Oxford University Press, Oxford, 2002.
 Müller, S., Mathematical problems in thin elastic sheets: scaling limits, packing, crumpling and singularities. Vector-valued partial differential equations and applications, 125–193, Lecture Notes in Math., 2179, Fond. CIME/CIME Found. Subser., Springer, Cham, 2017.
 S. Conti, A. Garroni, M. Ortiz: The line-tension approximation as the dilute limit of linear-elastic dislocations, Arch. Rational Mech. Anal., Vol. 218 (2015), no. 2, 699–755.